How to prove this $ \lim\limits_{n\to\infty} \frac{1}{n}\int_a ^b f'(x)\cos{(nx)} \mathrm{d}x = 0$? How to prove that the following is a limited function?
$$ \lim_{n\to\infty} \frac{1}{n}\int_a ^b f'(x)\cos{(nx)} \mathrm{d}x = 0$$
I tried to use inequality to prove that this is a limited function and I got nothing.
Thanks!
 A: $$|\cos{(nx)}|\leq 1\Rightarrow -1 \leq \cos{(nx)} \leq 1 $$ 


*

*When $f'(x) \geq 0:$ 
$$ -f'(x) \leq f'(x) \cos{(nx)} \leq f'(x) \Rightarrow \\ -\int_a^b f'(x) \leq \int_a^b f'(x) \cos{(nx)} \leq \int_a^b f'(x) \Rightarrow \\ -(f(b)-f(a))\leq \int_a^b f'(x) \cos{(nx)} \leq (f(b)-f(a) ) \Rightarrow \\ -(f(b)-f(a)) \frac{1}{n} \leq \frac{1}{n} \int_a^b f'(x) \cos{(nx)} \leq (f(b)-f(a) ) \frac{1}{n} $$


Taking the limit $n \rightarrow \infty$ we have the following:
$$\lim_{n \rightarrow \infty} [-(f(b)-f(a))] \frac{1}{n} \leq \lim_{n \rightarrow \infty}  \frac{1}{n} \int_a^b f'(x) \cos{(nx)} \leq \lim_{n \rightarrow \infty}  (f(b)-f(a) ) \frac{1}{n} \Rightarrow \\ 0 \leq \lim_{n \rightarrow \infty}  \frac{1}{n} \int_a^b f'(x) \cos{(nx)} \leq 0 \Rightarrow \\ \lim_{n \rightarrow \infty}  \frac{1}{n} \int_a^b f'(x) \cos{(nx)}=0$$ 


*

*When $f'(x) < 0:$ 
$$ f'(x) \leq f'(x) \cos{(nx)} \leq -f'(x) \Rightarrow \\
 \int_a^b f'(x) \leq \int_a^b f'(x) \cos{(nx)} \leq - \int_a^b f'(x) \Rightarrow \\ (f(b)-f(a))\leq \int_a^b f'(x) \cos{(nx)} \leq -(f(b)-f(a) ) \Rightarrow \\ (f(b)-f(a)) \frac{1}{n} \leq \frac{1}{n} \int_a^b f'(x) \cos{(nx)} \leq -(f(b)-f(a) ) \frac{1}{n} $$


Taking the limit $n \rightarrow \infty$ we have the following:
$$\lim_{n \rightarrow \infty} (f(b)-f(a)) \frac{1}{n} \leq \lim_{n \rightarrow \infty}  \frac{1}{n} \int_a^b f'(x) \cos{(nx)} \leq \lim_{n \rightarrow \infty}  [-(f(b)-f(a) ) ] \frac{1}{n} \Rightarrow \\ 0 \leq \lim_{n \rightarrow \infty}  \frac{1}{n} \int_a^b f'(x) \cos{(nx)} \leq 0 \Rightarrow \\ \lim_{n \rightarrow \infty}  \frac{1}{n} \int_a^b f'(x) \cos{(nx)}=0$$ 
A: Using integration by parts we have
$$\frac{1}{n} \int_a^b f'(x) \cos (nx) \mathrm{d}x=\frac{\cos (nx)}{n} f(x) \bigg]_{x=a}^{x=b}+\int_a^b f(x) \sin(nx) \mathrm{d}x. $$
You may finish off using the Riemann-Lebesgue lemma.
